The term is not מוֹצִיא שַׁבָּת (Motzi Shabbat) but rather מוֹצָאֵי שַׁבָּת (Motzaei Shabbat).
Both words comes from the shoresh י.צ.א. which has to do with going out. מוֹצִיא is a verb in the הפעיל (causative) binyan. It means "cause to go out" or more concisely, "bring out." I am not entirely sure of the grammatical form of מוֹצָאֵי but it appears to me to be a gerund: "the going out of Shabbat."
Motzaei Shabbat is the time when Shabbat "goes out."